7.6 Viscoelasticity and the energy principle

Two extensions close the chapter. The first admits time into the constitutive law: real materials are not perfectly elastic but viscoelastic, their stress depending on the rate of strain as well as its magnitude. The second recasts equilibrium as a minimum: the deformed shape of a loaded body is the one that minimises its total potential energy — the variational principle underlying the finite-element method.

Viscoelastic materials

A perfectly elastic solid stores all the work done on it and returns it on unloading; a viscous fluid dissipates it. Polymers, gels, biological tissues, and many soft materials do both: they store some energy and dissipate the rest, and their response depends on how fast they are loaded. Two spring-and-dashpot models capture the limiting behaviours, the spring supplying an elastic stress $\propto\varepsilon$ and the dashpot a viscous stress $\propto\dot\varepsilon$ .

Maxwell model — spring in series with a dashpot. Under a suddenly applied, sustained strain, the stress relaxes exponentially as the dashpot slowly gives way: $\sigma(t) = \sigma_0\,e^{-t/\tau}$ . Under a sustained stress, the strain grows without bound (viscous flow). It captures stress relaxation in materials that ultimately flow.
Kelvin–Voigt model — spring in parallel with a dashpot. Under a suddenly applied stress the dashpot resists the initial motion, and the strain creeps toward the elastic plateau $\varepsilon(\infty) = \sigma_0/k$ as $\varepsilon(t) = (\sigma_0/k)(1 - e^{-t/\tau})$ . It captures a solid that is delayed but not permanently flowing.

Both are usefully read in the frequency domain. Driven at angular frequency $\omega$ , a viscoelastic element presents a complex mechanical impedance

Z(\omega) \;=\; b \;+\; i\!\left(\omega m - \frac{k}{\omega}\right),

where

$Z(\omega)$: mechanical impedance (force per unit velocity) N·s/m
$b$: damping — the dashpot (dissipative) N·s/m
$k$: stiffness — the spring (elastic) N/m
$m$: inertial mass kg
$\omega$: drive angular frequency rad/s

whose real part $b$ measures dissipation and whose imaginary part sets the phase between force and motion. The stiffness dominates at low frequency, the mass at high, and the two cancel at resonance, where only the damping $b$ limits the response — the same impedance structure that governs any driven damped oscillator.

The principle of virtual work

Everything so far has been written as a differential equation — the Navier–Cauchy equation and its boundary conditions. There is an equivalent, and for computation more powerful, variational statement. The equilibrium configuration of an elastic body is the displacement field that minimises the total potential energy

\Pi[\mathbf{u}] \;=\; \int_V \tfrac12\,\sigma_{ij}\varepsilon_{ij}\,dV \;-\; \int_V f_i u_i\,dV \;-\; \int_S t_i u_i\,dS,

where

$\Pi[\mathbf{u}]$: total potential energy functional J
$\tfrac12\sigma_{ij}\varepsilon_{ij}$: stored elastic energy density J/m³
$f_i$: body force per unit volume N/m³
$t_i$: surface traction Pa

taken over all kinematically admissible displacement fields — those consistent with the imposed constraints. The first term is the strain energy stored in the deformation; the last two are the work done by the applied body and surface forces. Setting the first variation $\delta\Pi = 0$ recovers the Navier–Cauchy equation as its Euler–Lagrange equation, so the two formulations are equivalent.

The variational form is the foundation of the finite-element method. Rather than solve the differential equation directly, one restricts $\mathbf{u}$ to a finite family of shapes — piecewise-polynomial functions on a mesh of small elements — and minimises $\Pi$ over the finite set of parameters. Minimising a quadratic energy over a finite-dimensional space is a linear-algebra problem, and the solution converges to the true elastic field as the mesh is refined. Nearly every structural, mechanical, and biomechanical simulation in use rests on this principle.